lecture_35 - MA 265 LECTURE NOTES WEDNESDAY APRIL 9 Least...

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Least Squares Best Polynomial Approximation. We explain how to use the concept of a “least squares solution” to find a polynomial fit to a collection of data. Say that we have a list of data as follows: Time Amount t 1 y 1 t 2 y 2 . . . . . . t m y m We explain how to find a polynomial y ( t ) = a 0 + a 1 t + ··· + a n - 1 t n - 1 + a n t n of degree at most n which gives the best approximation to the data. Perform the following steps: #1. Form the inhomogeneous linear system a 0 + t 1 a 1 + ··· + t n - 1 1 a n - 1 + t n 1 a n = y 1 a 0 + t 2 a 1 + ··· + t n - 1 2 a n - 1 + t n 2 a n = y 2 . . . a 0 + t m a 1 + ··· + t n - 1 m a n - 1 + t n m a n = y m This involves m equations in ( n + 1) unknowns. We express this as the matrix product A x = b in terms of A = 1 t 1 ··· t n - 1 1 t n 1 1 t 2 ··· t n - 1 2 t n 2 . . . . . . . . . . . . . . . 1 t m ··· t n - 1 m t n m , x = a 0 a 1 . . . a n - 1 a n , and b = y 1 y 2 . . . y m . #2. Compute solution b x to the linear system ( A T A ) b x = ( A T b ) . Note that ( A T A ) is an ( n +1) × ( n +1) square, symmetric matrix – which does not depend on the number m of data points: A T A = " m X k =1 t i + j - 2 k # and A T b = " m X k =1 t j - 1 k y k # . #3. Form the best fit polynomial b y ( t ) = x 0 + x 1 t + ··· + x n - 1 t n - 1 + x n t n where b x = x 0 x 1 . . . x
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.

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lecture_35 - MA 265 LECTURE NOTES WEDNESDAY APRIL 9 Least...

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